Global Regularity for the Navier–Stokes Equations: A Geometric Resolution via the Evolutionary Layer-by-Layer Gauge Method

This paper presents a complete geometric construction of global smooth solutions for the three-dimensional Navier–Stokes equations, resolving the regularity problem as formulated by the Clay Mathematics Institute. The proof synthesizes two fun-damental ideas: the classification of degenerate solutions [20] and the continuous gauge method [21].The classification theorem shows that the only geometrically rigid degenerate case (rank 1, where streamlines are straight lines) cannot arise in solutions that are gauge-equivalent to a non-trivial axisymmetric flow. This eliminates the only potential obstruction to regularity. The continuous gauge method provides an analytical framework based on a modified entropy functional that controls both the velocity field and the gauge deformation. However, it does not specify how to construct the initial gauge diffeomorphism connecting an arbitrary field to an axisymmetric reference.The layer-by-layer construction fills this gap through two geometrically trans-parent steps:1. A time-dependent diffeomorphism that eliminates one spatial coordinate, re-ducing an arbitrary 3D flow to a 2D flow.2. A second time-dependent diffeomorphism that transforms the 2D flow into anaxisymmetric flow without swirl.Both steps are performed in an evolutionary manner, generating gauge fields that satisfy the equations of the continuous gauge method. Their composition provides the full evolutionary gauge for the original field. Using Fourier expansion in the angular variable, we prove that the class of fields admitting such an evolutionary gauge construction is dense. Uniform entropy estimates then show that this class is also closed, hence it coincides with the space of all smooth divergence-free fields. By the regularity of axisymmetric flows, every such initial data yields a globally regular solution.This work resolves the Navier–Stokes regularity problem in the affirmative, providing a constructive geometric proof that avoids the analytical difficulties that have hindered previous approaches.